I cant prove this. i dont understand what will be my hypothesis for induction.

Question

Consider the sequence $a_0, a_1, a_2,\dots$ of integers defined by $a_0 = 10$ and $a_n = 2a_{n-1}$, $n \geq 1$. Prove that $a_n = 2^n\cdot10$ for all $n \geq 0$.

Answer 1

Alternatively: $$a_n=2a_{n-1}=2(2a_{n-2})=2^2a_{n-2}=2^2(2a_{n-3})=2^3a_{n-3}=\cdots=2^na_0=2^n\cdot 10.$$

Answer 2

First of all $n = 1$ : $a_{1} = 2\cdot10 = 2^{1} \cdot 10$.

Step : $a_{n-1} = 2^{n-1}\cdot10$, so $2a_{n-1} = 2\cdot2^{n-1}\cdot10=2^{n}\cdot10 = a_{n}$.

Answer 3

Check for the base case: $a_0=2^0\cdot10=10$

Assume: $a_n=2^n\cdot10$

What we want to prove: $a_{n+1}=2^{n+1}\cdot10$

Now,

$a_{n+1}=2a_n=2\cdot(2^n\cdot10)=(2\cdot2^n)\cdot10=2^{n+1}\cdot10$